Strong Maximum Principle for Multi-Term Time-Fractional Diffusion Equations and its Application to an Inverse Source Problem

TL;DR

This paper establishes a strong maximum principle for multi-term time-fractional diffusion equations and applies it to prove the uniqueness of the inverse problem of determining the source term's temporal component.

Contribution

It extends the maximum principle from single-term to multi-term fractional diffusion equations and applies this to solve an inverse source identification problem.

Findings

Proved a strong maximum principle for multi-term fractional diffusion equations.

Established uniqueness in determining the source's temporal component.

Enhanced understanding of solution properties using multinomial Mittag-Leffler functions.

Abstract

In this paper, we establish a strong maximum principle for fractional diffusion equations with multiple Caputo derivatives in time, and investigate a related inverse problem of practical importance. Exploiting the solution properties and the involved multinomial Mittag-Leffler functions, we improve the weak maximum principle for the multi-term time-fractional diffusion equation to a stronger one, which is parallel to that for its single-term counterpart as expected. As a direct application, we prove the uniqueness for determining the temporal component of the source term with the help of the fractional Duhamel's principle for the multi-term case.